Methods and Applications for High-Frequency Biosignals Data

Lily Koffman

Department of Biostatistics, Johns Hopkins School of Public Health

Introduction: accelerometry data

Introduction: accelerometry data

Introduction: big accelerometry data

Outline

  • Can we identify someone from their walking pattern measured by a wrist-worn accelerometer? (Digital fingerprinting)
  • Can we identify someone from their walking pattern measured by a wrist-worn accelerometer in big, free-living datasets?
    • Can we accurately find walking and count steps in free-living datasets?
    • Can we generalize conclusions from free-living accelerometry data to the US population?
  • Can we apply these methods in other high-frequency data sources?

Outline

  • Can we identify someone from their walking pattern measured by a wrist-worn accelerometer? (Digital fingerprinting)
  • Can we identify someone from their walking pattern measured by a wrist-worn accelerometer in big, free-living datasets?
    • Can we accurately find walking and count steps in free-living datasets?
    • Can we generalize conclusions from free-living accelerometry data to the US population?
  • Can we apply these methods in other high-frequency data sources?

Can we identify someone from their walking pattern measured by a wrist-worn accelerometer?

Problem setup

Problem setup

Problem setup

Big picture method: time series to scalar predictors

Details of the method

For each second and each person:

  • Obtain joint distribution of acceleration and lag acceleration for a series of lags

  • Calculate scalar summaries of the joint distribution

  • I will walk through the process for one second, one person, and one lag

  • Intuition: walking is cyclic process. We want to leverage cyclic nature of walking.

Obtain joint distribution of acceleration and lag acceleration

Obtain joint distribution of acceleration and lag acceleration

Obtain joint distribution of acceleration and lag acceleration

Obtain joint distribution of acceleration and lag acceleration

Obtain joint distribution of acceleration and lag acceleration

Obtain joint distribution of acceleration and lag acceleration

Derive predictors from joint distribution

Derive predictors from joint distribution

Derive predictors from joint distribution

Derive predictors from joint distribution

Derive predictors from joint distribution

Repeat for multiple lags

Repeat for multiple seconds

Fingerprints summarize predictors for a given lag and are different across individuals

Fingerprints summarize predictors for a given lag and are different across individuals

Ready to fit models

Data

  • N = 32
  • 6 minutes of walking each
  • 3:1 train/test split

Karas et al. (2021)

Results

Koffman et al. (2023)

Why not use all possible lags?

Why not use all possible lags?

Functional regression approach

Hat tip to Edward Gunning for the idea for these figures

Why not use all possible lags?

Functional regression approach

Hat tip to Edward Gunning for the idea for these figures

Why not use all possible lags?

Functional regression approach

Hat tip to Edward Gunning for the idea for these figures

Why not use all possible lags?

Functional regression approach

Hat tip to Edward Gunning for the idea for these figures

Functional regression approach

Toy example: 4 observations per second, 2 seconds, 1 individual

\(v_j(s)\): \(s^{th}\) acceleration observation in second \(j\)

data \[\begin{bmatrix} v_1(1) & v_1(2) & v_1(3) & v_1(4) \\ v_2(1) & v_2(2) & v_2(3) & v_2(4) \\ \end{bmatrix} \]

Functional regression approach

Toy example: 4 observations per second, 2 seconds, 1 individual

\(v_j(s)\): \(s^{th}\) acceleration observation in second \(j\)

data \[\begin{bmatrix} v_1(1) & v_1(2) & v_1(3) & v_1(4) \\ v_2(1) & v_2(2) & v_2(3) & v_2(4) \\ \end{bmatrix} \]

acceleration matrix \[\begin{bmatrix} \color{blue}{v_1(2)} & \color{blue}{v_1(3)} & \color{blue}{v_1(4)} & \color{teal}{v_1(3)} & \color{teal}{v_1(4)} & \color{violet}{v_1(4)} \\ \color{blue}{v_2(2)} & \color{blue}{v_2(3)} & \color{blue}{v_2(4)} & \color{teal}{v_2(3)} & \color{teal}{v_2(4)} & \color{violet}{v_2(4)} \\ \end{bmatrix} \] lag acceleration matrix \[\begin{bmatrix} \color{blue}{v_1(1)} & \color{blue}{v_1(2)} & \color{blue}{v_1(3)} & \color{teal}{v_1(1)} & \color{teal}{v_1(2)} &\color{violet}{ v_1(1)} \\ \color{blue}{v_2(1)} & \color{blue}{v_2(2)} & \color{blue}{v_2(3)} & \color{teal}{v_2(1)} & \color{teal}{v_2(2)} &\color{violet}{ v_2(1)} \\ \end{bmatrix} \]

lag matrix \[\begin{bmatrix} \color{blue}{1} & \color{blue}{1} & \color{blue}{1} & \color{teal}{2} & \color{teal}{2} & \color{violet}{3}\\ \color{blue}{1} & \color{blue}{1} & \color{blue}{1} & \color{teal}{2} & \color{teal}{2} & \color{violet}{3}\\\end{bmatrix} \]

Number columns: \(4 \cdot (4-1) / 2 = 6\)

Functional regression: background

We’re familiar with predicting a (scalar) outcome using a (scalar) predictor

\[Y_i = \alpha + \beta X_i\]

Functional regression: background

Sometimes our outcome is a curve over time (i.e., function) rather than a single value.

We can model an outcome as a function of this curve:

\[Y_i = \alpha + \int X_i(t)\beta(t)dt + \epsilon_i\]

Functional regression: background

Sometimes our outcome is a curve over time (i.e., function) rather than a single value.

We can model an outcome as a function of this curve:

\[Y_i = \alpha + \int X_i(t)\beta(t)dt + \epsilon_i\]

Functional regression: background

Functional regression: background

Functional regression: background

Functional regression: background

  • Previous model took 1D function \(X(t)\) as predictor
  • Can generalize to more dimensions
  • Instead of a curve, the functional coefficient becomes a multivariate surface

Functional regression approach

Model outcomes as:

\[Y_{ij}^{i_0}\sim\text{Bernoulli}(p_{ij}^{i_0})\]

where \(Y_{ij}^{i_0} = 1\) if subject \(i\) in second \(j\) belongs to subject \(i_0\), and 0 otherwise

Functional regression approach

Model outcomes as:

\[Y_{ij}^{i_0}\sim\text{Bernoulli}(p_{ij}^{i_0})\]

where \(Y_{ij}^{i_0} = 1\) if subject \(i\) in second \(j\) belongs to subject \(i_0\), and 0 otherwise

Model:

\[\text{logit}(p_{ij}^{i_0}) =\beta_0^{i_0} + \int_{u=1}^S\int_{s=u}^SF_{i_0}\{ v_{ij}(s), v_{ij}(s-u), u\}dsdu \]

Functional regression approach

Model outcomes as:

\[Y_{ij}^{i_0}\sim\text{Bernoulli}(p_{ij}^{i_0})\]

where \(Y_{ij}^{i_0} = 1\) if subject \(i\) in second \(j\) belongs to subject \(i_0\), and 0 otherwise

Model:

\[\text{logit}(p_{ij}^{i_0}) =\beta_0^{i_0} + \int_{u=1}^S\int_{s=u}^SF_{i_0}\{ v_{ij}(s), v_{ij}(s-u), u\}dsdu \]

\(u = 1, \dots, S = 100\) (number of observations per second)

\(v_{ij}(s)\) = \(s^{th}\) for subject \(i\) in second \(j\)

\(F(\cdot, \cdot, \cdot)\): trivariate smooth function, takes values at every point in the domain of acceleration, lag acceleration, lag length

Functional regression approach

Model outcomes as:

\[Y_{ij}^{i_0}\sim\text{Bernoulli}(p_{ij}^{i_0})\]

where \(Y_{ij}^{i_0} = 1\) if subject \(i\) in second \(j\) belongs to subject \(i_0\), and 0 otherwise

Model:

\[\text{logit}(p_{ij}^{i_0}) =\beta_0^{i_0} + \int_{u=1}^S\int_{s=u}^SF_{i_0}\{ v_{ij}(s), v_{ij}(s-u), u\}dsdu \]

\(u = 1, \dots, S = 100\) (number of observations per second)

\(v_{ij}(s)\) = acceleration at centisecond \(s\) for subject \(i\) in second \(j\)

\(F(\cdot, \cdot, \cdot)\): trivariate smooth function, takes values at every point in the domain of acceleration, lag acceleration, lag length

Fit using penalized splines with a quadratic penalty on the functional coefficient (Wood 2016)

Functional regression approach: implementation

model = mgcv::gam(
  Y ~ te(
    accel_mat,
    lag_accel_mat,
    lag_mat,
    k = c(5, 5, 5),
    by = weight_mat),
  family = binomial(),
  method = "REML"
)

  • \(\texttt{te()}\): tensor product smooth

  • \(\texttt{k = c(5, 5, 5)}\) number of basis functions for each dimension of the tensor product smooth

  • \(\texttt{weight\_mat}\): matrix of weights of linear functionals of smooth terms. We use equal weights so the \(i,j^{\mathrm{th}}\) entry is \(\texttt{1/ncol(accel\_mat)}\)

  • \(\texttt{method="REML"}\): smoothing parameter selection with restricted maximum likelihood

Connection between two methods

  • Functional approach is generalization of grid cell approach
  • Instead of discretizing space into cells, relationship between acceleration, lag acceleration, lag length is modeled as smooth, continuous surface \(F\)
  • Grid cell approach: piecewise constant approximation of \(F\)
  • Both leverage same underlying structure: how acceleration, lag acceleration pairs characterize movement

New dataset

  • 153 individuals
  • 3 minutes of walking each
  • Two sessions at least one week apart

ZJU-GaitAcc (Zhang et al. 2015)

Functional regression results

Rank-1 (rank-5) % accuracies

  • Train and test on session 1
    • Logistic regression: 92 (97)
    • Functional regression: 98 (100)
  • Train on session 1, test on session 2
    • Logistic regression: 41 (75)
    • Functional regression: 53 (69)

Why not use fancier models?

Rank-1 (rank-5) % accuracies

  • Train and test on session 1
    • Logistic regression: 92 (97)
    • Functional regression: 98 (100)
    • XGBoost: 93 (99)
  • Train on session 1, test on session 2
    • Logistic regression: 41 (75)
    • Functional regression: 53 (69)
    • XGBoost: 58 (78)

Koffman, Crainiceanu, and Leroux (2024)

Outline

  • Can we identify someone from their walking pattern measured by a wrist-worn accelerometer? (Digital fingerprinting)
    \(\rightarrow\) Yes!
  • Can we identify someone from their walking pattern measured by a wrist-worn accelerometer in big, free-living datasets?
    • Can we accurately find walking and count steps in free-living datasets?
    • Can we generalize conclusions from free-living accelerometry data to the US population?
  • Can we apply these methods in other high-frequency data sources?

Outline

  • Can we identify someone from their walking pattern measured by a wrist-worn accelerometer? (Digital fingerprinting)
    \(\rightarrow\) Yes!
  • Can we identify someone from their walking pattern measured by a wrist-worn accelerometer in big, free-living datasets?
    • Can we accurately find walking and count steps in free-living datasets?
    • Can we generalize conclusions from free-living accelerometry data to the US population?
  • Can we apply these methods in other high-frequency data sources?

Outline

  • Can we identify someone from their walking pattern measured by a wrist-worn accelerometer? (Digital fingerprinting)
    \(\rightarrow\) Yes!
  • Can we identify someone from their walking pattern measured by a wrist-worn accelerometer in big, free-living datasets?
    • Can we accurately find walking and count steps in free-living datasets?
    • Can we generalize conclusions from free-living accelerometry data to the US population?
  • Can we apply these methods in other high-frequency data sources?

Can we accurately find walking and count steps in free-living datasets?

Validation: identifying walking in free-living datasets

  • Search for free-living datasets with wrist-worn accelerometry and gold standard step counts
    • Free-living \(\neq\) lab
  • Implement open-source algorithms for step counting from wrist accelerometry
  • Evaluate performance of algorithms

Capturing gold standard steps in one dataset

Validation: identifying walking in free-living datasets

Open-source algorithms and 3 datasets with gold-standard step counts

Koffman and Muschelli (2024)

Application: idenfiying walking and counting steps in NHANES

  • \(>15,000\) participants
  • \(7\) days of wrist accelerometry
  • \(10\)Tb of data
  • Over 1 year computation time
  • Open source processing pipeline
  • Publicly available data repository
  • First nationally representative estimate of steps in the US population using open source algorithms
Koffman and Muschelli (2025a)

Application: idenfiying walking and counting steps in NHANES

How many average daily steps do Americans take?

Application: idenfiying walking and counting steps in NHANES

Do estimates differ by algorithm?

Application: idenfiying walking and counting steps in NHANES

Are more steps associated with lower mortality risk?

Koffman, Crainiceanu, and Muschelli (2024)

Outline

  • Can we identify someone from their walking pattern measured by a wrist-worn accelerometer? (Digital fingerprinting)
    \(\rightarrow\) Yes!
  • Can we identify someone from their walking pattern measured by a wrist-worn accelerometer in big, free-living datasets?
    • Can we accurately find walking and count steps in free-living datasets?
      \(\rightarrow\) Yes!
    • Can we generalize conclusions from free-living accelerometry data to the US population?
  • Can we apply these methods in other high-frequency data sources?

Outline

  • Can we identify someone from their walking pattern measured by a wrist-worn accelerometer? (Digital fingerprinting)
    \(\rightarrow\) Yes!
  • Can we identify someone from their walking pattern measured by a wrist-worn accelerometer in big, free-living datasets?
    • Can we accurately find walking and count steps in free-living datasets?
      \(\rightarrow\) Yes!
    • Can we generalize conclusions from free-living accelerometry data to the US population?
  • Can we apply these methods in other high-frequency data sources?

Can we generalize conclusions from free-living accelerometry data to the US population?

Sex differences in steps?

Do males take more steps than females? At what points during the day?

Function on scalar regression

  • Outcome: steps profile over the course of the day (function)

  • Predictors: age, sex (scalars)

  • Model: \[\mathbb{E}[\mathrm{steps}_i(s)] = \beta_0(s) + \beta_1(s)\mathbb{I}\{\mathrm{sex}_i=\mathrm{female}\} + \beta_2(s)(\mathrm{age}_i-50) \] \(i\): participant; \(s \in \{1, \dots, 1440\}\): each minute of the day

  • Interpretation

    • \(\beta_0(s)\): expected steps for reference sex category and age 50
    • \(\beta_1(s)\): change in steps for females compared to males, holding age constant
    • Can be interpreted at specific points of the day (e.g., how many more steps do males take at 12pm?)

Function on scalar regression: implementation

  • Fast univariate inference (FUI) (Cui et al. 2021)

  • Fit separate (univariate) GLM at each point \(s\), smooth the resulting point estimates

  • Bootstrap subjects to get confidence bands

Function on scalar regression: implementation

  • Fast univariate inference (FUI) (Cui et al. 2021)

  • Fit separate (univariate) GLM at each point \(s\), smooth the resulting point estimates

  • Bootstrap subjects to get confidence bands

  • BUT: NHANES is not a simple random sample

  • Individuals sampled in geographic clusters, minority groups oversampled

  • Are estimates valid for population-level inference?

NHANES Sample Design Tutorial

Survey function on scalar regression

  • No method exists for survey-weighted function on scalar regression

  • Standard regression, well developed methods and software to take into accounts weights and correlation between clusters (e.g. \(\texttt{svyglm}\), \(\texttt{svycoxph}\)) (Lumley 2010)

  • FUI built on separate GLMS

  • Idea: incorporate survey weights into the GLMs and use survey-aware replication/bootstrap methods for inference

Survey function on scalar regression: simulation

First simulation study to evaluate function on scalar regression in complex survey settings

  • Generate superpopulation
    • Each individual has functional outcome and covariate
    • Individuals belong to clusters (geographic areas)
    • Functional outcomes correlated within clusters
  • Sample from superpopulation in each iteration of simulation
    • Selection probability related to functional outcome (informative sampling)
  • Compare:
    • Unweighted model w/ standard bootstrap
    • Weighted models with weighted bootstrap
    • Weighted models with complex survey bootstrap (2)
  • Evaluate:
    • Bias in coefficient estimation
    • Coverage of confidence intervals

Survey function on scalar regression: simulation

Koffman et al. (2025)

Survey function on scalar regression: software

model_fit = svyfosr::svyfui(steps_mat ~ age + sex,
                            weights = survey_weight,
                            data = steps_df,
                            family = gaussian(),
                            boot_type = "BRR",
                            num_boots = 500,
                            parallel = TRUE,
                            seed = 2213)
Koffman and Muschelli (2025b)

Survey function on scalar regression: application

Complex survey function on scalar regression: application

Outline

  • Can we identify someone from their walking pattern measured by a wrist-worn accelerometer? (Digital fingerprinting)
    \(\rightarrow\) Yes!
  • Can we identify someone from their walking pattern measured by a wrist-worn accelerometer in big, free-living datasets?
    • Can we accurately find walking and count steps in free-living datasets?
      \(\rightarrow\) Yes!
    • Can we generalize conclusions from free-living accelerometry data to the US population?
      \(\rightarrow\) Yes!
  • Can we apply these methods in other high-frequency data sources?

Outline

  • Can we identify someone from their walking pattern measured by a wrist-worn accelerometer? (Digital fingerprinting)
    \(\rightarrow\) Yes!
  • Can we identify someone from their walking pattern measured by a wrist-worn accelerometer in big, free-living datasets?
    • Can we accurately find walking and count steps in free-living datasets?
      \(\rightarrow\) Yes!
    • Can we generalize conclusions from free-living accelerometry data to the US population?
      \(\rightarrow\) Yes!
  • Can we apply these methods in other high-frequency data sources?

Can we identify someone from their walking pattern measured by a wrist-worn accelerometer in big, free-living datasets?

Digital fingerprinting in NHANES

  • Use highly specific method to identify walking in NHANES (minimize false positives) (Karas et al. 2019)
  • \(N = 13{,}000\) individuals with 3 minutes walking per person
  • 3:1 train/test split
  • Logistic regression + weighting to overcome class imbalance
  • 43% rank-1 accuracy
  • 73% rank-5 accuracy
  • 97% rank-1% accuracy (correct subject is in the top 130 predictions)
  • 100% rank-5% accuracy (correct subject is in the top 650 predictions)

Digital fingerprinting in NHANES

Koffman, Muschelli, and Crainiceanu (2025)

Outline

  • Can we identify someone from their walking pattern measured by a wrist-worn accelerometer? (Digital fingerprinting)
    \(\rightarrow\) Yes!
  • Can we identify someone from their walking pattern measured by a wrist-worn accelerometer in big, free-living datasets?
    \(\rightarrow\) Yes!
    • Can we accurately find walking and count steps in free-living datasets?
      \(\rightarrow\) Yes!
    • Can we generalize conclusions from free-living accelerometry data to the US population?
      \(\rightarrow\) Yes!
  • Can we apply these methods in other high-frequency data sources?

Outline

  • Can we identify someone from their walking pattern measured by a wrist-worn accelerometer? (Digital fingerprinting)
    \(\rightarrow\) Yes!
  • Can we identify someone from their walking pattern measured by a wrist-worn accelerometer in big, free-living datasets?
    \(\rightarrow\) Yes!
    • Can we accurately find walking and count steps in free-living datasets?
      \(\rightarrow\) Yes!
    • Can we generalize conclusions from free-living accelerometry data to the US population?
      \(\rightarrow\) Yes!
  • Can we apply these methods in other high-frequency data sources?

Can we apply these methods in other high-frequency data sources?

Hemodynamics and cardiac surgery

  • Pipeline to analyze minute-level multivariate hemodynamics data during cardiac surgery (Koffman 2025)
  • Methods to model relationship between hemodynamics and acute kidney injury (Goeddel et al. 2025)

Arterial waveform

Arterial waveform

Arterial waveform

Fingerprinting with arterial waveform

Fingerprinting with arterial waveform

  • Obtain predictors for many different lags and cut points
  • Use predictors that are top 10 contributors to first 30 PCs (\(\approx\) 100 predictors)
  • Fit XGBoost model on 727 patients
  • Mean (SD) 7 (1.8) minutes per patient, range 3-16 minutes

Fingerprinting with arterial waveform

Outline

  • Can we identify someone from their walking pattern measured by a wrist-worn accelerometer? (Digital fingerprinting)
    \(\rightarrow\) Yes!
  • Can we identify someone from their walking pattern measured by a wrist-worn accelerometer in big, free-living datasets?
    \(\rightarrow\) Yes!
    • Can we accurately find walking and count steps in free-living datasets?
      \(\rightarrow\) Yes!
    • Can we generalize conclusions from free-living accelerometry data to the US population?
      \(\rightarrow\) Yes!
  • Can we apply these methods in other high-frequency data sources?
    \(\rightarrow\) Yes!

Future Directions

  • Changes in fingerprint (both walking and waveform) to predict changes in function
  • Designing real-time interventions based on arterial waveform during surgery
  • Extending survey function on scalar regression to longitudinal outcomes
  • Standardizing processing and analysis pipelines for wearable accelerometry

Thank you!



References

Cui, Erjia, Andrew Leroux, Ekaterina Smirnova, and Ciprian M. Crainiceanu. 2021. “Fast Univariate Inference for Longitudinal Functional Models.” Journal of Computational and Graphical Statistics 31 (1): 219–30. https://doi.org/10.1080/10618600.2021.1950006.
Goeddel, Lee A., Marina Hernandez, Lily Koffman, Zachary Murphy, Ashish K. Khanna, Michael Robich, Glenn Whitman, et al. 2025. “Fine-Mapping the Association of Acute Kidney Injury With Mean Arterial and Central Venous Pressures During Coronary Artery Bypass Surgery.” Anesthesia &Amp; Analgesia 140 (6): 1439–49. https://doi.org/10.1213/ane.0000000000007500.
Karas, Marta, Marcin Stra̧czkiewicz, William Fadel, Jaroslaw Harezlak, Ciprian M Crainiceanu, and Jacek K Urbanek. 2019. “Adaptive Empirical Pattern Transformation (ADEPT) with Application to Walking Stride Segmentation.” Biostatistics 22 (2): 331–47. https://doi.org/10.1093/biostatistics/kxz033.
Karas, Marta, Jacek Urbanek, Ciprian Crainiceanu, Jaroslaw Harezlak, and William Fadel. 2021. “Labeled Raw Accelerometry Data Captured During Walking, Stair Climbing and Driving.” PhysioNet. https://doi.org/10.13026/51H0-A262.
Koffman, Lily. 2025. bloodlines: Visualize hemodynamic time series data. https://github.com/jhuwit/bloodlines.
Koffman, Lily, Ciprian Crainiceanu, and Andrew Leroux. 2024. “Walking Fingerprinting.” Journal of the Royal Statistical Society Series C: Applied Statistics 73 (5): 1221–41. https://doi.org/10.1093/jrsssc/qlae033.
Koffman, Lily, Ciprian Crainiceanu, and John Muschelli. 2024. “Comparing Step Counting Algorithms for High-Resolution Wrist Accelerometry Data in NHANES 2011–2014.” Medicine & Science in Sports & Exercise 57 (4): 746–55. https://doi.org/10.1249/mss.0000000000003616.
Koffman, Lily, Sunan Gao, Xinkai Zhou, Andrew Leroux, Ciprian Crainiceanu, and John Muschelli III. 2025. “Function on Scalar Regression with Complex Survey Designs.” https://arxiv.org/abs/2511.05487.
Koffman, Lily, and John Muschelli. 2024. “Evaluating Step Counting Algorithms on Subsecond Wrist-Worn Accelerometry: A Comparison Using Publicly Available Data Sets.” Journal for the Measurement of Physical Behaviour 7 (1). https://doi.org/10.1123/jmpb.2024-0008.
———. 2025a. “Minute level step counts and physical activity data from the National Health and Nutrition Examination Survey (NHANES) 2011-2014.” PhysioNet. https://doi.org/10.13026/9N0R-TV02.
———. 2025b. svyfosr: Survey-Weighted Function on Scalar Regression. https://github.com/jhuwit/svyfosr.
Koffman, Lily, John Muschelli, and Ciprian Crainiceanu. 2025. “Walking Fingerprinting Using Wrist Accelerometry During Activities of Daily Living in NHANES.” https://arxiv.org/abs/2506.17160.
Koffman, Lily, Yan Zhang, Jaroslaw Harezlak, Ciprian Crainiceanu, and Andrew Leroux. 2023. “Fingerprinting Walking Using Wrist-Worn Accelerometers.” Gait & Posture 103 (June): 92–98. https://doi.org/10.1016/j.gaitpost.2023.05.001.
Lumley, Thomas. 2010. Complex Surveys: A Guide to Analysis Using R: A Guide to Analysis Using R. John Wiley; Sons.
Wood, Simon N. 2016. “P-Splines with Derivative Based Penalties and Tensor Product Smoothing of Unevenly Distributed Data.” Statistics and Computing 27 (4): 985–89. https://doi.org/10.1007/s11222-016-9666-x.
Zhang, Yuting, Gang Pan, Kui Jia, Minlong Lu, Yueming Wang, and Zhaohui Wu. 2015. “Accelerometer-Based Gait Recognition by Sparse Representation of Signature Points With Clusters.” IEEE Transactions on Cybernetics 45 (9): 1864–75. https://doi.org/10.1109/TCYB.2014.2361287.